Determinants of Elementary Matrices

Recall that an elementary matrix $E$ arises from performing exactly one of the following elementary row operations on $I$:

• 1. Multiplying a row by a constant $k$ where $k ≠ 0$ ($kR_{a} \to R_{a}$).

• 2. Adding (or subtracting) a multiple $k$ of a row to another ($R_a + kR_b \to R_a$).

• 3. Interchanging two rows ($R_a \leftrightarrow R_b$).

We will now look at some techniques in evaluating the determinants of these elementary matrices.

Determinants of Elementary Matrices from Multiplying a Row by a Constant

If $E$ results from multiplying a single row of $I$ by a constant $k$, it follows that $\det(E) = k$. For example, consider the following elementary matrix has $\det(E) = 5$.

(1)

\begin{align} E = \begin{bmatrix} 1 & 0 & 0\\ 0 & 5 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}

Determinants of Elementary Matrices by Adding/Subtracting a Multiple of One Row to Another

If $E$ results from adding or subtracting a multiple of one row of $I$ to another, then it follows that $\det(E) = 1$. For example, the following elementary matrix has $\det(E) = 1$.

(2)

\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 3 & 0 & 1 \end{bmatrix}

Determinants of Elementary Matrices by Interchanging Two Rows

If $E$ results from interchanging two rows of $I$, then it follows that $\det(E) = -1$. For example, the following elementary matrix has $\det(E) = -1$.

(3)

\begin{bmatrix} 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{bmatrix}